Question

Barb is going to cover a rectangular area 8 feet by 10 feet with rectangular paving blocks that are 4 inches by 8 inches by 2 inches to make a flat pati. What is the minimum number of paving blocks she will need if all the paving blocks will face the same direction? (Note: Barb will not cut any of the paving blocks.) A. 80 B. 360 C. 601 D. 960 E. \(1,213\)

Short answer

Answer: 360 paving blocks

Step-by-step solution

Convert units of measurements

First, we need to convert the dimensions of the paving blocks to feet so we can easily compare them to the dimensions of the rectangular area. The dimensions of the paving block are 4 inches by 8 inches by 2 inches. There are 12 inches in one foot, so we can convert the dimensions as follows: 1. \(4 inches \times \frac{1 foot}{12 inches} = \frac{4}{12} = \frac{1}{3} feet\) 2. \(8 inches \times \frac{1 foot}{12 inches} = \frac{8}{12} = \frac{2}{3} feet\) 3. \(2 inches \times \frac{1 foot}{12 inches} = \frac{2}{12} = \frac{1}{6} feet\) So the dimensions of the paving block in feet are \(\frac{1}{3} feet \times \frac{2}{3} feet \times \frac{1}{6} feet\).

Determine the number of blocks for the length of the rectangular area

Since the paving blocks need to face the same direction, we will place the longest side (8 inches or \(\frac{2}{3} feet\)) of the blocks along the length of the area. Next, we need to determine how many paving blocks can fit within the length (8 feet) of the area. Divide the length of the area (8 feet) by the length of the paving block (\(\frac{2}{3} feet\)): \(8 feet ÷ \frac{2}{3} feet = \frac{8}{1} \times \frac{3}{2} =8 \times 1.5 =12\) So, 12 paving blocks fit within the length of the rectangular area.

Determine the number of blocks for the width of the rectangular area

Now, we need to determine how many paving blocks can fit within the width (10 feet) of the area. To do this, we divide the width of the area (10 feet) by the width of the paving block (\(\frac{1}{3} feet\)): \(10 feet ÷ \frac{1}{3} feet = \frac{10}{1} \times \frac{3}{1} = 10 \times 3 = 30\) So, 30 paving blocks fit within the width of the rectangular area.

Calculate the minimum number of paving blocks required

Now that we know how many blocks fit within the length and width of the rectangular area, we can calculate the total number of paving blocks required: Minimum number of paving blocks = \((12 \text{ blocks} \times 30 \text{ blocks}) = 360 \text{ blocks}\) So the minimum number of paving blocks required is 360, which corresponds to option B.

Key concepts

Unit Conversion

Understanding unit conversion is essential for solving many math and real-life problems. In the context of the ACT Math Prep, converting units from inches to feet is a commonly tested skill. The process involves dividing by 12 because there are 12 inches in a foot.

For instance, a paving block that is 4 inches long is equivalent to \(\frac{4}{12}\) or \(\frac{1}{3}\) feet when converted. Similarly, 8 inches becomes \(\frac{8}{12}\) or \(\frac{2}{3}\) feet, and 2 inches translates to \(\frac{2}{12}\) or \(\frac{1}{6}\) feet. Through this simple division by 12, we can seamlessly compare and combine measurements that were initially in different units.

Rectangular Area Calculation

Area calculation is a fundamental concept in geometry, and it becomes particularly relevant when dealing with rectilinear shapes like rectangles. The area of a rectangle is calculated by multiplying its length by its width. This simple formula is invaluable when determining how much material is needed to cover a space, like when paving a patio.

For the ACT Math Prep, understanding this can help solve problems that require spatial reasoning. For example, calculating how many 4-inch by 8-inch blocks are needed to cover an 8-foot by 10-foot area requires knowledge of both unit conversion and area calculation. Once units are consistent (usually converted to the same unit), we multiply the number of blocks that fit along the length by the number of blocks that fit along the width to find the total amount necessary to cover the area.

Paving Blocks Problem Solving

Solving paving block problems involves a mix of unit conversion and area calculation, as well as strategic thinking. The goal is to determine the number of blocks required to cover a given area without cutting any blocks. To tackle such problems:

• Ensure all measurements are in the same unit.
• Calculate how many blocks fit along the length and width of the area, considering the blocks' orientation.
• Multiply the two numbers to find the total number of blocks required.

In the example provided, aligning the long side of the paving block with the length of the area is a strategic decision. With this approach, we can calculate that precisely 12 blocks fit along one dimension, and 30 along the other, leading to a requirement of 360 blocks. This clear methodology ensures accuracy and efficiency in solving similar practical problems.